Adam Lallana (L) scored the winner with the last kick of the game.
Despite Slovakia having Martin Skrtel sent off in the 57th minute, Allardyce looked set to become the first England manager since Bobby Robson in 1982 not to win his first game in charge.
But in the fifth minute of stoppage time in Trnava, Liverpool midfielder Lallana squeezed a shot between Slovakia goalkeeper Matus Kozacik’s legs to get England up and running in Group F.
“It was pretty nerve-racking at the end because the 10 men scenario meant we must win,” Allardyce told ITV.
“We got what we deserved in the end and won in the last minute. We have dominated the game and got a very important win.”
Allardyce, nicknamed ‘Big Sam’, succeeded Roy Hodgson after England’s humiliating Euro 2016 elimination by Iceland.
Kozacik’s gaffe enabled him to become the ninth England manager to start his tenure with a win.
Lallana’s goal, on a clammy night at City Arena, spared England from a repeat of the result from their group-stage encounter with Slovakia at Euro 2016.
It also meant that captain Wayne Rooney finished as a winner on a night when he won his 116th cap to become England’s most-capped outfield player.
Allardyce had vowed that Rooney would no longer play as a number 10 on his watch, but the captain nonetheless lined up in midfield beside Jordan Henderson in a 4-1-4-1 system.
With Raheem Sterling hugging the touchline on the right and Lallana flitting infield from the left, it left lone striker Harry Kane isolated.
It took England until mid-way through the first half to present Kane with anything approaching a shooting chance and when Kyle Walker picked him out at the near post, the number nine missed his kick.
Slovakia gave England a scare when Michal Duris robbed Danny Rose and flashed the ball across the six-yard box, narrowly eluding Dusan Svento.
The first half ended with Sterling drilling a shot wide, but England were toiling.
Allardyce shuffled his pack at half-time, switching Sterling and Lallana and pushing Rooney upfield, but it took the dismissal of Skrtel for the visitors to take the upper hand.
Booked in the first half for catching Kane with a flailing arm, the former Liverpool centre-back saw red after crudely stamping on the same opponent’s calf as he knelt on the pitch.
Rooney registered England’s first shot on target in the 65th minute, a weak curler that Kozacik pushed away uncertainly, before Sterling almost caught the Slovakia goalkeeper out with an improvised chip.
Allardyce sent on Dele Alli and Theo Walcott for Henderson and Sterling, while Daniel Sturridge replaced Kane late on.
It looked as if the visitors would come no closer than a left-foot snapshot from Lallana that hit the left-hand post and bounced across goal.
The former Southampton captain continued to look like England’s best chance of a goal, with one shot parried by Kozacik and another fizzing wide.
Walcott found the net in the 90th minute, tucking the ball away after Sturridge’s shot was blocked, only for his celebrations to be cut short by an offside flag.
But just as victory appeared destined to elude England, Lallana gathered Rose’s cross and unleashed a hopeful shot that crept between Kozacik’s legs and dribbed into the net at the near post.
“Unfortunately, Martin Skrtel got sent off and then it was only about our heroic defending with the support of our fantastic fans,” said Slovakia coach Jan Kozak.
“This is football. You concede a goal in the 95th minute. We almost held as great a team as England to two successive draws.”